Re: A Differential Equation

Re: A Differential Equation

Originally Posted by Bruno J.

What are your reasons for believing it's true? Or is that a secret?

no, it's not a secret! it should be true because it is consistent with something which is expected to be true about the structure of centralizers in the second complex Weyl algebras. i'm not giving more details because it's a very long story!

Re: A Differential Equation

NonCommAlg means the free -ring on 3 indeterminates, modded out by the relations provided.

It's the ring of complex polynomials over , except without commutivity except as those relations dictate.

The partial derivative, I assume, is just a formalization of the analytic notion of a partial derivative. It's the same as the partial derivative for an ordinary polynomial over in x, y, and z, except that the non-commutivity now has to be respected.

Thus means an element of A, a funky non-commuting polynomial in x, y, z, that isn't just a constant value. It must include at least some power of x, y, or z - at least, after those relations are taken into account (meaning yx-xy is not in , because under these relations, it's a constant polynomial).

The is required because otherwise b=1 (actually, any b in ) trivially works.